Vol.40 (2018) — 10 articles found.

Certain Properties of Tracial Approximation ${\rm C^*}$-Algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (4) 2018, pp. 104-133
Vol.40 (4) 2018
George A. Elliott, FRSC; Qingzhai Fan; Xiaochun Fang Details
(Received: 2019-04-07 )
(Received: 2019-04-07 )

George A. Elliott, FRSC,Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu,

Qingzhai Fan,Department of Mathematics, Shanghai Maritime University, Shanghai, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang,Department of Mathematics, Tongji University, Shangha, China 200092; e-mail: xfang@mail.tongji.edu.cn

Abstract/Résumé:

We show that the following properties of the \({\rm C^*}\)-algebras in a class \(\Omega\) are inherited by simple unital \({\rm C^*}\)-algebras in the class \({\rm TA}\Omega\): \((1)\) \(\beta\)-comparison (\(1\leq \beta < \infty\)), \((2)\) \(n\)-comparison, \((3)\) trace \(\mathcal{Z}\)– absorption, \((4)\) \(m\)-almost divisibility, \((5)\) \((n,m) ~(m\neq 0)\) comparison, and \((6)\) tracial approximate divisibility. As an application, every unital simple \({\rm C^*}\)-algebra with tracial topological rank at most \(k\) has the property of \(k\)-comparison. Also as an application, let \(A\) be an infinite-dimensional simple unital \({\rm C^*}\)-algebra such that \(A\) has one of the above-listed properties. Suppose that \(\alpha: G\to {\rm Aut}(A)\) is an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then the crossed product \({\rm C^*}\)-algebra \({\rm C^*}( G, A,\alpha)\) also has the property under consideration.

On considère plusieurs propriétés d’une C*-algèbre simple à élément unité qui sont héritées par approximation traciale. Comme application on démontre que ces propriétés sont aussi héritées par la C*-algèbre produit croisé associée à une action d’un groupe fini qui possède la propriété de Rokhlin traciale.

Keywords: C*-algebra, Cuntz Semigroup, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Certain Properties of Tracial Approximation ${C^*}$-Algebras

Uniqueness of the Index Map in Real K-theory

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (4) 2018, pp. 101-103
Vol.40 (4) 2018
Ralf Meyer Details
(Received: 2019-03-15 , Revised: 2019-04-01 )
(Received: 2019-03-15 , Revised: 2019-04-01 )

Ralf Meyer,Mathematisches Institut, Georg-August Universitaet Goettingen, Bunsenstrasse 3-5, 37073 Goettingen, Germany; email: rmeyer2@uni-goettingen.de

Abstract/Résumé:

The index map in topological K-theory for real Banach algebra extensions is a natural transformation from the first K-theory of the quotient to the zeroth K-theory of the ideal. We show that any such natural transformation is an integer multiple of the index map.

L’application index dans la K-théorie des extensions d’algèbres de Banach réelles est une transformation naturelle entre le premier K-groupe de l’algèbre quotient et le zéroième K-groupe de l’idéal. On démontre qu’une telle transformation naturelle doit être un multiple intégral de l’application index.

Keywords: Index map, real K-theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Real K-theory

Uniqueness of the Index Map in Banach Algebra K-theory, II

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (3) 2018, pp. 91-100
Vol.40 (3) 2018
George A. Elliott Details
(Received: 2018-09-01 , Revised: 2018-09-01 )
(Received: 2018-09-01 , Revised: 2018-09-01 )

George A. Elliott,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in the theory of real Banach algebras is unique as a natural transformation, up to an integral multiple, and modulo a (unique) two-torsion “ghost” map arising from the order-two K\(_1\)-group of the Banach algebra \({\mathbb R}\) (of real numbers). (In the earlier paper this was shown for complex Banach algebras, of course without the “ghost” map, but in way—using Bott periodicity to pass to the opposite parity—that is not available for real Banach algebras. The present approach yields a new proof in the complex case.)

On démontre que l’application index dans la K-théorie des algèbres de Banach réelles (ou complexes) est essentiellment unique.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-theory, II

A Survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (3) 2018, pp. 65-90
Vol.40 (3) 2018
Frederic Latremoliere Details
(Received: 2018-05-05 , Revised: 2018-05-05 )
(Received: 2018-05-05 , Revised: 2018-05-05 )

Frederic Latremoliere,Department of Mathematics, University of Denver, Denver CO 80208; e-mail: frederic@math.du.edu

Abstract/Résumé:

We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.

Nous étudions les propriétés de conservation des symmétries des espaces quantiques pour la proximité duale, sous des conditions naturelles d’équicontinuité et de non dégénérescence. Ces propriétés sont exprimées naturellement dans le language de la proximité covariante, qui permet de discuter la convergence d’actions de groupes et semigroupes sur les espaces quantiques. Nous explorons le problème de la convergence des suites de Cauchy pour la proximité covariante, qui capture, grâce a un théoreme de compacité, le fait que les actions de monoides propres passent à la limite pour la proximité duale.

Keywords: C*-dynamical systems, Gromov-Hausdor convergence, Gromov-Hausdor distance for proper monoids, Lip-norms, Monge- Kantorovich distance, Noncommutative metric geometry, proper monoids, quantum metric spaces

AMS Subject Classification: K-theory and operator algebras -including cyclic theory, Noncommutative geometry (__ la Connes) 46L80, 58B34

PDF(click to download): A Survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity

Lusternik-Schnirelmann Category in Commutative Algebra and The Homotopy Lie Algebra

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (2) 2018, pp. 61-64
Vol.40 (2) 2018
Benjamin Briggs Details
(Received: 2018-05-30 , Revised: 2018-05-30 )
(Received: 2018-05-30 , Revised: 2018-05-30 )

Benjamin Briggs,Department of Mathematics, University of Toronto, Ontario, Canada M5S 2E4; e-mail: ben.briggs@mail.utoronto.ca

Abstract/Résumé:

Here we continue the development of Lusternik-Schnirelmann category in local commutative algebra. In particular, we present a version of Félix-Halperin’s mapping theorem which is valid in any characteristic. Then we briefly discuss some consequences for the behaviour of the homotopy Lie algebra of a local homomorphism. This is a short announcement of some of the results in the author’s thesis.

Nous continuons le développement de la catégorie deLusternik-Schnirelmann en algèbre commutative locale. En particulier, nous présentons une version du théorème de Félix-Halperin qui est valable pour toute caractéristique. Nous tirerons ensuite brièvement des conséquences sur le comportement de l’algèbre de Lie d’homotopie d’un homomorphisme local. Ceci constitue une courte annonce de certains des résultats contenus dans la thèse de l’auteur.

Keywords: Lusternik-Schnirelmann category, dg algebras, the homotopy Lie algebra

AMS Subject Classification: Syzygies and resolutions, Differential graded algebras and applications, Rational homotopy theory 13D02, 16E45, 55P62

PDF(click to download): Lusternik-Schnirelmann Category in Commutative Algebra and The Homotopy Lie Algebra

The Rieffel Projection Via Groupoids

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (2) 2018, pp. 55-60
Vol.40 (2) 2018
George A. Elliott; Dickson Wong Details
(Received: 2018-05-05 , Revised: 2018-05-05 )
(Received: 2018-05-05 , Revised: 2018-05-05 )

George A. Elliott,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Dickson Wong,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: dickson.wong@mail.utoronto.ca

Abstract/Résumé:

An elementary groupoid construction is shown to underlie Rieffel’s Hilbert module construction of a non-trivial projection in the irrational rotation C*-algebra.

Une construction élémentaire de groupoïde se révèle à la base de la construction de Rieffel à module de Hilbert d’un projecteur non-trivial dans la C*-algèbre d’une rotation irrationnelle.

Keywords: Irrational rotation algebra, Rieffel projection, groupoid construction

AMS Subject Classification: General theory of $C^*$-algebras, K-theory and operator algebras -including cyclic theory, Noncommutative topology 46L05, 46L80, 46L85

PDF(click to download): The Rieffel Projection Via Groupoids

A Positive-Definite Energy Functional for Axially Symmetric Maxwell’s Equations on Kerr-de Sitter Black Hole Spacetimes

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (2) 2018, pp. 39-54
Vol.40 (2) 2018
Nishanth Gudapati Details
(Received: 2017-11-04 , Revised: 2018-04-06 )
(Received: 2017-11-04 , Revised: 2018-04-06 )

Nishanth Gudapati,Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT-06511, USA; e-mail: nishanth.gudapati@yale.edu

Abstract/Résumé:

We prove that there exists a phase space of canonical variables, for the initial value problem for axially symmetric Maxwell fields with compactly supported initial data and propagating in Kerr-de Sitter black hole spacetimes, such that their motion is restricted to the level sets of a positive-definite Hamiltonian, despite the ergo-region.

On démontre qu’il existe un espace de phase de variables canoniques, pour le problème des valeurs initials pour les champs de Maxwell symétriques à donneés initiales de support compact et à propagation dans les espaces-temps de trou noir Kerr-de Sitter, tel que leur motion est restrainte aux ensembles de niveau d’une hamiltonienne de type positif, en dépit de l’ergo-région.

Keywords: Kerr-de Sitter Black Holes, Stability of Black Holes, Wave Maps

AMS Subject Classification: Conservation laws, Electromagnetic fields 35L65, 83C50

PDF(click to download): A Positive-Definite Energy Functional for Axially Symmetric Maxwell's Equations on Kerr-de Sitter Black Hole Spacetimes

Perfect Powers that are Sums of Consecutive Squares

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (2) 2018, pp. 33-38
Vol.40 (2) 2018
Vandita Patel Details
(Received: 2017-08-08 , Revised: 2017-09-11 )
(Received: 2017-08-08 , Revised: 2017-09-11 )

Vandita Patel,Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Room 6290, Toronto, Ontario, Canada M5S 2E4; e-mail: vandita@math.utoronto.ca

Abstract/Résumé:

We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.

Nous déterminons toutes les puisances \(n\)-ièmes qui peuvent être écrits comme la somme d’au plus 10 carrés consécutifs.

Keywords: Exponential equation

AMS Subject Classification: Exponential equations 11D61

PDF(click to download): Perfect Powers that are Sums of Consecutive Squares

A Note on the Lipschitz Selection

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (1) 2018, pp. 29-32
Vol.40 (1) 2018
Alexander Brudnyi Details
(Received: 2018-01-02 , Revised: 2018-01-10 )
(Received: 2018-01-02 , Revised: 2018-01-10 )

Alexander Brudnyi,Department of Mathematics and Statistics, University of Calgary, Alberta, Canada
T2N 1N4; e-mail: abrudnyi@ucalgary.ca

Abstract/Résumé:

We present an alternate proof of the passage from the finiteness principle for metric trees to the construction of the core in the C. Fefferman and Shvartsman finiteness theorem for Lipschitz selection problems.

On présente une preuve alternative du passage du principe de la finitude pour les arbres métriques jusqu’à la construction du noyau dans le théorème de finitude de C. Fefferman et Shvartsman pour la sélection des problèmes Lipschitz.

Keywords: Lipschitz selection, length space, metric tree, regular covering

AMS Subject Classification: Sobolev spaces and other spaces of ``smooth'' functions; embedding theorems; trace theorems 46E35

PDF(click to download): A Note on the Lipschitz Selection

Almost Periodicity in Time of Solutions of the Toda Lattice

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (1) 2018, pp. 1-28
Vol.40 (1) 2018
Ilia Binder; David Damanik; Milivoje Lukic; Tom VandenBoom Details
(Received: 2017-11-11 , Revised: 2017-12-21 )
(Received: 2017-11-11 , Revised: 2017-12-21 )

Ilia Binder,Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Toronto, Ontario, Canada M5S 2E4;
e-mail: ilia@math.toronto.edu

David Damanik,Department of Mathematics, Rice University, Houston, TX 77005, USA; e-mail: damanik@rice.edu

Milivoje Lukic,Department of Mathematics, Rice University, Houston, TX 77005, USA; e-mail: milivoje.lukic@rice.edu

Tom VandenBoom,Department of Mathematics, Rice University, Houston, TX 77005, USA; e-mail: tvandenboom@rice.edu

Abstract/Résumé:

We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and show the boundedness and almost periodicity in time and space of solutions.

On étudie un problème de Cauchy pour le système Toda avec des conditions initiales presque périodiques. On considère des conditions initiales pour lesquelles l’opérateur de Jacobi associé est absolument continu et a un spectre qui satisfait à une condition à la façon de Craig, afin de montrer que les solutions sont bornées et presque périodiques dans la variable spatiale aussi que temporelle.

Keywords: Toda lattice, almost periodic Jacobi operators

AMS Subject Classification: Almost periodic solutions, Completely integrable systems; integrability tests; bi-Hamiltonian structures; hierarchies (KdV; KP; Toda; etc.), Jacobi (tridiagonal) operators (matrices) and generalizations 35B15, 37K10, 47B36

PDF(click to download): Almost Periodicity in Time of Solutions of the Toda Lattice

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